Properties of the Exponential Function - Maple Help

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Exploring properties of the Exponential function

 Main Concept The graph of the exponential function $f\left(x\right)={ⅇ}^{x}$ has a very interesting property. If you draw a vertical line (green in the graph to the right) from a point $\left({x}_{0},{ⅇ}^{{x}_{0}}\right)$ on the graph down to the point $\left({x}_{0},0\right)$ on the x-axis, then draw another line 1 unit to the left (red), to the point $\left({x}_{0}-1,0\right)$, and then finally complete the triangle by drawing the line through $\left({x}_{0}-1,0\right)$ and $\left({x}_{0},{ⅇ}^{{x}_{0}}\right)$ (magenta), this final line will just touch the graph of ${ⅇ}^{x}$ at this latter point without passing through the graph; that is, this line is tangent to the graph of ${ⅇ}^{x}$ at the point $\left({x}_{0},{ⅇ}^{{x}_{0}}\right)$. This property (with the base of the triangle having length 1) and the specification that the function has value 1 at $x=0$ completely and uniquely determine the exponential function ${ⅇ}^{x}$.   The slope of this (magenta) line tells you how fast the function is growing. So this property of the exponential function can be summarized this way: At every point, how big the exponential function is and how fast it is growing are the same.

 Click or drag on the graph to see a demonstration of this property.   



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